Here, we have that for \(c_{s*} \ll 1\)  the spectral tilt is such that
it will turn blue when \(\beta > 15/14 \simeq 1.07\) , while for
\(c_{s*}=1\) , the spectral tilt is blue already for \(\beta > 0.9\) . As
in this case we still have \(\beta \simeq 4.33\)  (for \(c_{s*}=1\) ), thus,
for large \(Q\)  the spectrum will always be quite blue tilted. This
agrees with the previous results for this type of WI
model [1]}, [2]}, which have shown that
this dissipation coefficient, within the momonial chaotic inflaton
potential models, can only be consistent with the observational
dataFrom the Planck Collaboration [3]},
the result for the spectral tilt is \(n_s=0.9658\pm 0.0040\)  (95\(\%\) 
CL, Planck TT,TE,EE+lowE+lensing+BK15+BAO+running) at pivot scale
\(k_*=0.05\,{\rm Mpc}^{-1}\) . when \(Q_* \ll 1\) , i.e., in the weak
dissipation regime of WI. In this regime of \(Q_* \ll 1\) , the growing
mode is not an issue, since \(G(Q_*) \rightarrow 1\) . As in the case (a) of a
linear in the temperature dissipation coefficient, this model can
benefit of a DBIWI realization by exploring the effect of a small
\(c_s\)  value, which will help to suppress the growing mode by
decreasing \(\beta \) . However, this is expected to only happen for a
smaller value for \(c_s\)  than the one shown for the case (a), \(c_{s*}^2\lesssim 0.2\) . This case also illustrates well the importance in
deriving the full dependence of the dissipation coefficient in both
the temperature and in the inflaton field amplitude. Despite cases (b)
and (c) have exactly the same temperature dependence, the fact that
the dissipation coefficient in (c) has an explicit dependence on
\(\phi \) , it implies that an agreement with the observational value for
the tilt of the scalar power spectrum can only be achieved in the weak
dissipation regime of WI (for the standard case of \(c_{s*}=1\) ), while
in (b) we can robustly go in the strong dissipation regime of WI, even
with the presence of the growing mode, which in that case is benign.
